To solve this problem, we need to break it down into two parts: determining how long the ball was in the air and calculating its speed as it left the table. We can use the principles of physics, particularly the equations of motion under the influence of gravity, to find our answers.
Time in the Air
First, let's find out how long the ball was in the air. The height of the table is given as 4.23 feet. We can use the formula for the time of free fall, which is derived from the second equation of motion:
Here, h is the height (4.23 ft), g is the acceleration due to gravity (approximately 32.2 ft/s²), and t is the time in seconds. Rearranging the formula to solve for t gives us:
Now, substituting the values:
- t = √(2 * 4.23 ft / 32.2 ft/s²)
Calculating this step-by-step:
- 2 * 4.23 = 8.46 ft
- 8.46 ft / 32.2 ft/s² ≈ 0.262 ft/s²
- t = √(0.262) ≈ 0.512 seconds
Result for Time
The ball was in the air for approximately 0.512 seconds.
Calculating the Speed at Departure
Next, we need to find the speed of the ball as it left the table. Since the ball rolls off horizontally, its initial vertical speed is 0. The horizontal distance it traveled is 5.11 feet. We can use the formula for horizontal motion:
Where d is the horizontal distance (5.11 ft), v is the horizontal speed, and t is the time in the air (0.512 seconds). Rearranging this formula to solve for v gives us:
Substituting the known values:
- v = 5.11 ft / 0.512 seconds
Calculating this gives:
Final Speed Result
The speed of the ball at the instant it left the table was approximately 9.96 ft/s.
Summary
In summary, the ball was in the air for about 0.512 seconds and had a horizontal speed of approximately 9.96 ft/s when it rolled off the table. This problem illustrates the principles of projectile motion, where the horizontal and vertical motions can be analyzed separately.
